As an exhaust gas apparatus of an internal combustion engine to be used by an automotive vehicle, there is known an exhaust gas apparatus as shown in FIG. 19 (for example Patent Document 1). In FIG. 19, the known exhaust gas apparatus 4 is adapted to allow an exhaust gas to be introduced therein after the exhaust gas exhausted from an engine 1 serving as an internal combustion engine passes through an exhaust manifold 2 and is purified by a catalytic converter 3.
The exhaust gas apparatus 4 is constituted by a front pipe 5 connected to the catalytic converter 3, a center pipe 6 connected to the front pipe 5, a main muffler 7 connected to the center pipe 6 and serving as a sound deadening device, a tail pipe 8 connected to the main muffler 7, and a sub-muffler 9 connected to the tail pipe 8.
As shown in FIG. 20, the main muffler 7 has an expansion chamber 7a for expanding and introducing therein the exhaust gas through small holes 6a formed in the center pipe 6, and a resonance chamber 7b held in communication with a downstream opened end 6b of the center pipe 6, so that the exhaust gas introduced into the resonance chamber 7b from the downstream opened end 6b of the center pipe 6 can have an exhaust sound muted with a specified frequency due to Helmholtz resonator effect.
Here, if the pipe length of the projection portion of the center pipe 6 projecting into the resonance chamber 7b is L1(m), the cross sectional area of the center pipe 6 is S(m2), the volume of the resonance chamber 7b is V(m3), and the velocity of sound in air is c(m/s), the resonance frequency fn(Hz) in the air can be obtained by a following equation (1) in regard to the Helmholtz resonator effect.
                              f          n                =                              c                          2              ⁢                                                          ⁢              π                                ⁢                                    S                                                L                  1                                ·                V                                                                        (        1        )            
As apparent from the equation (1), the resonance frequency can be tuned to a low frequency side by making large the volume V of the resonance chamber 7b or otherwise by making long the pipe length L1 of the projection portion of the center pipe 6 while can be tuned to a high frequency side by making small the volume V of the resonance chamber 7b or otherwise by making short the pipe length L1 of the projection portion of the center pipe 6.
The sub-muffler 9 is adapted to suppress the sound pressure from being increased with the column air resonance generated in response to the pipe length of the tail pipe 8 in the tail pipe 8 by the pulsation of the exhaust gas during the operation of the engine 1.
In general, the tail pipe 8 having an upper stream opened end 8a and a lower stream opened end 8b at the respective upstream and downstream sides of the exhaustion direction of the exhaust gas is subjected to incident waves caused by the pulsation of the exhaust gas during the operation of the engine 1 at the upper stream opened end 8a and the lower stream opened end 8b, thereby generating an air column resonance with a wavelength. The air column resonance has a basic component of a frequency with a half wavelength equal to the pipe length L of the tail pipe 8, and has frequencies several times higher than that of the half wavelength.
More specifically, the wavelength λ1 of the air column resonance of a basic vibration (primary component) is roughly double the pipe length L of the tail pipe 8, while the wavelength λ2 of the air column resonance of the secondary component is roughly one time the pipe length L of the tail pipe 8. The wavelength λ3 of the air column resonance of the third component is ⅔ times the pipe length L of the tail pipe 8. Therefore, the tail pipe 8 has therein standing waves having respective nodes of sound pressures at the upper stream opened end 8a and the lower stream opened end 8b. 
The column air resonance frequency fa can be represented by a following equation (2).
                    fa        =                              c                          2              ⁢                                                          ⁢              L                                ⁢          n                                    (        2        )            
Here, “c” represents the velocity of sound (m/s), “L” represents the pipe length of the tail pipe (m), and “n” represents a harmonic degree. As apparent from the equation (2), the velocity of sound “c” has a constant value responsive to an ambient temperature. The longer the pipe length L of the tail pipe 8 becomes, nearer the air column frequency “fa” moves to the low frequency side, thereby making it easy to give rise to a noise problem caused by the air column resonance of the exhaust sound in the low frequency area.
For example, if the velocity of sound “c” is 400 m/s, the primary component “f1” and the secondary component “f2” of the exhaust gas sound by the air column resonance respectively become 166.7 Hz and 333.3 Hz in the case of the pipe length “L” of the tail pipe 8 being 1.2 m. On the other hand, the primary component “f1” and the secondary component “f2” of the exhaust gas sound by the air column resonance respectively become 66.7 Hz and 133.3 Hz in the case of the pipe length “L” of the tail pipe 8 being 3.0 m. It is therefore understood that the longer the pipe length L of the tail pipe 8 becomes, nearer the air column frequency “fa” moves to the low frequency side.
The frequency “fe(Hz)” of the exhaust gas pulsation of the engine 1 is represented by a following equation (3).
                    fe        =                              Ne            60                    ×                      N            2                                              (        3        )            
Here, “Ne” is an engine speed (rpm), and “N” is a number of cylinders of the engine (natural number).
The sound pressure level (dB) of the exhaust gas sound becomes remarkably high in the primary component “f1” of the exhaust gas by the air column resonance generated in response to a specified engine speed “Ne”. Further, the sound pressure level (dB) of the exhaust gas sound also becomes remarkably high in the secondary component “f2”.
For example, if the velocity of sound “c” is 400 m/s, and the number “N” of the cylinder is set at “4” for the 4-cylider engine, there is caused an air column resonance having a primary component “f1” of the frequency 66.7 Hz when the engine speed “Ne” becomes 2000 rpm, while another air column resonance having a secondary component “f2” of the frequency 133.3 Hz is caused when the engine speed “Ne” becomes 4,000 rpm in the case of the pipe length “L” of the tail pipe 8 being 3.0 m.
Especially in the case that the air column resonance is generated in the low frequency area below 100 Hz of the frequency of the exhaust gas pulsation of the engine 1, there is caused a problem in sound. For example when the air column resonance is generated in the tail pipe 8 at a low engine speed of 2000 rpm, the exhaust gas sound is transmitted to the passenger room of the vehicle, thereby leading to generation of a muffled sound and thus to giving an unpleasant feeling to a driver.
For this reason, there is provided a sub-muffler 9 smaller in volume than the main muffler 7 at the optimum position of the tail pipe 8 with respect to an antinode portion having a high sound pressure of a standing wave generated by the air column resonance, thereby preventing the air column resonance from being generated.
Therefore, for example if the sound velocity “c” is 400 m/s, and the pipe length “L” of the tail pipe 8 is 3.0 m with no sub-muffler 9, there is caused an air column resonance below 100 Hz of the frequency of the exhaust gas pulsation of the engine 1 (below 3,000 rpm of the engine speed “Ne”) as previously mentioned. In contrast, if the sub-muffler 9 is supported on the tail pipe 8, and the pipe length “L” of the tail pipe 8 extending rearwardly of the sub-muffler 9 is 1.5 m, the primary component “f1” of the exhaust gas sound by the air column resonance is 133.3 Hz, and the engine speed “Ne” is 4,000 rpm, thereby leading to causing the air column frequency fa to move to the high frequency side.
For this reason, the sub-muffler 9 supported on the tail pipe 8 can suppress the muffled sound in the passenger room at the low speed, viz., 2000 rpm of the rotation speed of the engine 1, thereby preventing an unpleasant feeling from being given to the driver.
On the other hand, it is considered to reduce the production cost and the weight of the exhaust gas apparatus 4 by eliminating the previously mentioned sub-muffler 9. As one of the measures, it is considered to tune the resonance frequency of the main muffler 7 connected to the upper stream opened end 8a of the tail pipe 8 with the frequency of the air column resonance to mute the exhaust gas sound of the air column resonance of the tail pipe 8 in the resonance chamber of the main muffler 7.
More specifically, it may be considered that in accordance with the equation (1), the volume “V” of the resonance chamber 7b is expanded, or the length L1 of the projection portion of the center pipe 6 is lengthened to conduct the tuning of the resonance frequency of the resonance chamber 7b toward the low frequency side, thereby preliminarily muting in the resonance chamber 7b the air column resonance generated in the tail pipe 8.